For federated authentication and authorisation to work, IdPs and SPs have to agree about many things: trust, attribute vocabularies, locations of services, key material, etc. Federations are a largely administrative arrangement to save every SP from having to negotiate agreements with every IdP they want to use.
Federations typically have rules about what particular data items mean, how reliable IdP data needs to be, what SPs should (and shouldn't) do with it, etc. To join the federation, members must agree to abide by the rules.
The UK Access Management Federation for Education and Research, commonly 'the UK Federation', has been established jointly by the Joint Information Systems Committee (JISC) on behalf of FE, and BECTA on behalf of schools. Similar federations are being established in the US, Europe, and elsewhere.